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Mirrors > Home > MPE Home > Th. List > tgbtwntriv2 | Structured version Visualization version GIF version |
Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwntriv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwntriv2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
tgbtwntriv2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 771 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝑥)) | |
2 | tkgeom.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
3 | tkgeom.d | . . . . . 6 ⊢ − = (dist‘𝐺) | |
4 | tkgeom.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tkgeom.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | 5 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐺 ∈ TarskiG) |
7 | tgbtwntriv2.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | 7 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 ∈ 𝑃) |
9 | simplr 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝑥 ∈ 𝑃) | |
10 | simpr 488 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → (𝐵 − 𝑥) = (𝐵 − 𝐵)) | |
11 | 2, 3, 4, 6, 8, 9, 8, 10 | axtgcgrid 26412 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 = 𝑥) |
12 | 11 | adantrl 716 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 = 𝑥) |
13 | 12 | oveq2d 7189 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥)) |
14 | 1, 13 | eleqtrrd 2837 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵)) |
15 | tgbtwntriv2.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
16 | 2, 3, 4, 5, 15, 7, 7, 7 | axtgsegcon 26413 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) |
17 | 14, 16 | r19.29a 3200 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 distcds 16680 TarskiGcstrkg 26379 Itvcitv 26385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-nul 5175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-iota 6298 df-fv 6348 df-ov 7176 df-trkgc 26397 df-trkgcb 26399 df-trkg 26402 |
This theorem is referenced by: tgbtwncom 26437 tgbtwntriv1 26440 tgcolg 26503 legid 26536 hlid 26558 lnhl 26564 tglinerflx2 26583 mirreu3 26603 mirconn 26627 symquadlem 26638 outpasch 26704 hlpasch 26705 |
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